Theorem qsqeqor 11087

Description: The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.)

Assertion

Ref	Expression
qsqeqor	⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))

Proof of Theorem qsqeqor

Step	Hyp	Ref	Expression
1		qre 10025	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
2	1	ad3antrrr 496	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐴 ∈ ℝ)
3		simplr 533	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐴)
4		qre 10025	. . . . . . 7 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ)
5	4	ad3antlr 497	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
6		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
7		sq11 11049	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
8	2, 3, 5, 6, 7	syl22anc 1279	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
9		orc 724	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))
10	8, 9	biimtrdi 163	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
11		oveq1 6092	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2))
12	11	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2)))
13		oveq1 6092	. . . . . . . . 9 ⊢ (𝐴 = -𝐵 → (𝐴↑2) = (-𝐵↑2))
14	13	adantl 277	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (-𝐵↑2))
15		qcn 10034	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
16		sqneg 11035	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℂ → (-𝐵↑2) = (𝐵↑2))
17	15, 16	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → (-𝐵↑2) = (𝐵↑2))
18	17	ad2antlr 493	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (-𝐵↑2) = (𝐵↑2))
19	14, 18	eqtrd 2271	. . . . . . 7 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2))
20	19	ex 115	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = -𝐵 → (𝐴↑2) = (𝐵↑2)))
21	12, 20	jaod 729	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
22	21	ad2antrr 492	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
23	10, 22	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
24	17	eqeq2d 2250	. . . . . 6 ⊢ (𝐵 ∈ ℚ → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
25	24	ad3antlr 497	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
26	1	ad3antrrr 496	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℝ)
27		simplr 533	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ 𝐴)
28		qnegcl 10036	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
29		qre 10025	. . . . . . . . 9 ⊢ (-𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
30	28, 29	syl 14	. . . . . . . 8 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
31	30	ad3antlr 497	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
32		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
33	4	le0neg1d 8845	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
34	33	ad3antlr 497	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
35	32, 34	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
36		sq11 11049	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
37	26, 27, 31, 35, 36	syl22anc 1279	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
38		olc 723	. . . . . 6 ⊢ (𝐴 = -𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))
39	37, 38	biimtrdi 163	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
40	25, 39	sylbird 170	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
41	21	ad2antrr 492	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
42	40, 41	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
43		0z 9655	. . . . . 6 ⊢ 0 ∈ ℤ
44		zq 10026	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
45	43, 44	ax-mp 5	. . . . 5 ⊢ 0 ∈ ℚ
46		qletric 10676	. . . . 5 ⊢ ((0 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
47	45, 46	mpan 428	. . . 4 ⊢ (𝐵 ∈ ℚ → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
48	47	ad2antlr 493	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
49	23, 42, 48	mpjaodan 810	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
50		qnegcl 10036	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ)
51		qre 10025	. . . . . . . . . 10 ⊢ (-𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
52	50, 51	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
53	52	ad3antrrr 496	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → -𝐴 ∈ ℝ)
54		simplr 533	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐴 ≤ 0)
55	1	le0neg1d 8845	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
56	55	ad3antrrr 496	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
57	54, 56	mpbid 147	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ -𝐴)
58	4	ad3antlr 497	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
59		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
60		sq11 11049	. . . . . . . 8 ⊢ (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
61	53, 57, 58, 59, 60	syl22anc 1279	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
62	61	biimpd 144	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) → -𝐴 = 𝐵))
63		qcn 10034	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
64		sqneg 11035	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))
65	63, 64	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → (-𝐴↑2) = (𝐴↑2))
66	65	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴↑2) = (𝐴↑2))
67	66	eqeq1d 2247	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
68	67	ad2antrr 492	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
69		negcon1 8578	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
70	63, 15, 69	syl2an 289	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
71		eqcom 2240	. . . . . . . 8 ⊢ (-𝐵 = 𝐴 ↔ 𝐴 = -𝐵)
72	70, 71	bitrdi 196	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵 ↔ 𝐴 = -𝐵))
73	72	ad2antrr 492	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (-𝐴 = 𝐵 ↔ 𝐴 = -𝐵))
74	62, 68, 73	3imtr3d 202	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → 𝐴 = -𝐵))
75	74, 38	syl6 33	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
76	21	ad2antrr 492	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
77	75, 76	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
78	52	ad3antrrr 496	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐴 ∈ ℝ)
79		simplr 533	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ≤ 0)
80	55	ad3antrrr 496	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
81	79, 80	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐴)
82	30	ad3antlr 497	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
83		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
84	33	ad3antlr 497	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
85	83, 84	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
86		sq11 11049	. . . . . . 7 ⊢ (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
87	78, 81, 82, 85, 86	syl22anc 1279	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
88	65, 17	eqeqan12d 2254	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
89	88	ad2antrr 492	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
90	63	ad3antrrr 496	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℂ)
91	15	ad3antlr 497	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ∈ ℂ)
92	90, 91	neg11ad 8633	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵))
93	87, 89, 92	3bitr3d 218	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
94	93, 9	biimtrdi 163	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
95	21	ad2antrr 492	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
96	94, 95	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
97	47	ad2antlr 493	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
98	77, 96, 97	mpjaodan 810	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
99		qletric 10676	. . . 4 ⊢ ((0 ∈ ℚ ∧ 𝐴 ∈ ℚ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
100	45, 99	mpan 428	. . 3 ⊢ (𝐴 ∈ ℚ → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
101	100	adantr 276	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
102	49, 98, 101	mpjaodan 810	1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209   class class class wbr 4130  (class class class)co 6085  ℂcc 8177  ℝcr 8178  0cc0 8179   ≤ cle 8361  -cneg 8498  2c2 9355  ℤcz 9644  ℚcq 10019  ↑cexp 10975

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-seqfrec 10885  df-exp 10976

This theorem is used by:  4sqlem10  13166